Linear algebra is the study of vectors and linear transformations. This text is suitable for a sophomore level linear algebra course taught in about twenty-five lectures. It is designed both for engineering and science majors, but has enough abstraction to be useful for potential math majors. Our goal in writing it was to produce students who can perform computations with linear systems and also understand the concepts behind these computations.
- Linear algebra is the study of vectors and linear transformations.
- The two key properties of vectors are that they can be added together and multiplied by scalars, so we make the following definition.
- Matrices are a powerful tool for calculations involving linear transformations. It is important to understand how to find the matrix of a linear transformation and properties of matrices.
- This section address some fundamental questions of vector spaces: Subspaces - When is a subset of a vector space itself a vector space? Linear Independence - Given a collection of vectors, is there a way to tell whether they are independent, or if one is a "linear combination'' of the others? Dimension - Is there a consistent definition of how "big'' a vector space is? Basis - How do we label vectors? Can we write any vector as a sum of some basic set of vectors?
- Given a linear transformation, it is highly desirable to write its matrix with respect to a basis of eigenvectors. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. A square matrix that is not diagonalizable is called defective.
- You may have noticed that we have only rarely used the dot product. That is because many of the results we have obtained do not require a preferred notion of lengths of vectors. Once a dot or inner product is available, lengths of and angles between vectors can be measured--very powerful machinery and results are available in this case.
- Symmetric matrices have many applications.